Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Show that if there is no harvest \(\left(C_{t}=0\right)\) and both \(N_{t}\) and \(N_{t-8}\) are equal to \(N_{*}\) then \(N_{t+1}=N_{*}\) If we divide all terms of Equation 8.14 by \(N_{*},\) we get $$ \frac{N_{t+1}}{N_{*}}=0.94 \frac{N_{t}}{N_{*}}+\frac{N_{t-8}}{N_{*}}\left[0.06+0.0567\left\\{1-\left(\frac{N_{t-8}}{N_{*}}\right)^{2.39}\right\\}\right]-0.94 \frac{C_{t}}{N_{*}} $$ We might define new variables, \(D_{t}=\frac{N_{t}}{N_{*}}\) and \(E_{t}=\frac{C_{t}}{N_{*}}\) and have the equation $$ D_{t+1}=0.94 D_{t}+D_{t-8}\left[0.06+0.0567\left\\{1-D_{t-8}^{2.39}\right\\}\right]-0.94 E_{t} $$ Equation 8.15 is simpler by one parameter \(\left(N_{*}\right)\) than Equation 8.14 and yet illustrates the same dynamical properties. Rather than use new variables, it is customary to simply rewrite Equation 8.14 with new interpretations of \(N_{t}\) and \(C_{t}\) and obtain $$ N_{t+1}=0.94 N_{t}+N_{t-8}\left[0.06+0.0567\left\\{1-N_{t-8}^{2.39}\right\\}\right]-0.94 C_{t} $$ \(N_{t}\) now is a fraction of \(N_{*},\) the number supported without harvest, and \(C_{t}\) is a fraction of \(N_{*}\) that is harvested. Equation 8.14 says that the population in year \(t+1\) is affected by three things: the number of female whales in the previous year \(\left(N_{t}\right),\) the "recruitment" of eight year old female whales into the population subject to harvest, and the harvest during the previous year \(\left(C_{t}\right)\)

Step-by-Step Solution

Answer

If there is no harvest and both \(N_t = N_*\) and \(N_{t-8} = N_*\), then \(N_{t+1} = N_*\).

Step 1: Understand Given Conditions

We are asked to examine the dynamics of the population when there is no harvest \(C_t = 0\) and both \(N_t = N_*\) and \(N_{t-8} = N_*\). This implies \(E_t = 0\) and \(D_t = 1\) since \(D_t = \frac{N_t}{N_*}\) and \(E_t = \frac{C_t}{N_*}\).

Step 2: Set Up the Equation with Given Conditions

Substitute \(D_t = 1\), \(D_{t-8} = 1\), and \(E_t = 0\) into the equation \[D_{t+1} = 0.94 D_t + D_{t-8} [0.06 + 0.0567 (1 - D_{t-8}^{2.39})] - 0.94 E_t\].

Step 3: Simplify the Expression

Given \(D_t = 1\) and \(D_{t-8} = 1\), the equation becomes \[D_{t+1} = 0.94 \times 1 + 1 \times [0.06 + 0.0567 (1 - 1^{2.39})] - 0.94 \times 0\].

Step 4: Calculate Inner Terms

Evaluate \(1^{2.39}\), which is 1. Therefore, \[1 - 1^{2.39} = 0\]. Thus, the equation simplifies to \[D_{t+1} = 0.94 \times 1 + 1 \times 0.06 + 1 \times 0.0567 \times 0\].

Step 5: Finalize the Calculation

The equation now becomes: \[D_{t+1} = 0.94 + 0.06 = 1.00\]. Thus, \(D_{t+1} = 1\), which implies \(N_{t+1} = N_*\) because \(D_{t+1} = \frac{N_{t+1}}{N_*} = 1\).

Key Concepts

Mathematical Biology

Mathematical biology is a fascinating and growing field that uses mathematical techniques to model biological processes. The goal is to understand and predict complex behaviors that occur in biological systems. This can range from modeling how diseases spread, to understanding the dynamics of ecosystems, like whale populations subjected to hunting.
Often, mathematical models in biology simplify real-world scenarios to make them more manageable while still capturing essential biological interactions. In our exercise, for example, simplifications are made by expressing whale populations as fractions of an environment's carrying capacity represented by \(N_*\).
The models can help scientists and ecologists make informed decisions on conservation and management, especially when precise long-term predictions are needed. By applying mathematical biology principles, students learn to solve real-world problems using abstract and analytical approaches.

Difference Equations

Difference equations are a type of mathematical equation used to model changes over discrete time steps. They are particularly useful in scenarios where changes occur at regular intervals, such as annually or monthly.
In our exercise, we use a difference equation to model whale population dynamics over time. The equation includes terms for current and past population sizes and any influencing factors like harvesting.

• The main variables are expressed as fractions of the carrying capacity \(N_*\). For example, \(D_t\) represents the current population size as a fraction of \(N_*\).
• Difference equations help in predicting future population sizes based on current and past data points.

The key to understanding difference equations is recognizing patterns and relationships between time intervals, which helps predict future trends.

Population Dynamics

Population dynamics is the study of how populations of living organisms change over time and what factors influence these changes. It is a fundamental aspect of understanding ecosystems and managing species conservation.
The exercise describes how whale populations evolve without external harvest intervention. The variables \(N_t\) and \(N_{t-8}\) play crucial roles in showing how the population maintains equilibrium when no harvesting occurs. In the absence of harvest \(C_t = 0\), population stability was demonstrated with \(N_{t+1} = N_*\).

• Understanding population dynamics involves examining recruitment rates, natural mortality, and self-regulating factors like competition for resources.
• Models can demonstrate sustainability through equilibrium states where populations neither grow nor decline significantly within certain bounds.

By mastering population dynamics, conservation efforts can become more strategic, ensuring the future viability of species like the whales in our model.